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Physical review. B, Condensed matter

Polaron density matrix and effective mass at finite temperature

Ashok Sethia, Fumio Hirata, Yoshitaka Tanimura, Yashwant Singh

Published Sep 1, 1999• 6 citations
Publisher

Abstract

We calculate the density matrix elements of the polaron using Feynman's variational method. The density matrix is diagonalized, and the eigenvalues and eigenfunctions are derived. These results and the calculated density matrix are used to evaluate the root-mean-square (rms) displacements ${R}_{\ensuremath{\beta}}$ and the effective mass (EM) ${m}^{*}$ for various values of coupling strengths between electron and medium and of the temperature. We find that EM is related to the rms displacement. The temperature dependence of the EM is in qualitative agreement with experiment.

DOI
10.1103/physrevb.60.7245

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Adequacy of various definitions of polaron effective mass ${m}^{\mathrm{*}}$ is examined. The expression for ${m}^{\mathrm{*}}$ obtained by using the zero-temperature kernel is compared with the zero-temperature limit of the temperature-dependent effective mass obtained by employing the finite-temperature kernel. We find that consistent results are obtained when Saitoh's definition is employed. Our calculations are illustrated for a simple case of weak electron-phonon interaction.

Physical review. B, Condensed matter•Feb 15, 1988•19 citations
We calculate the density matrix elements of the polaron using Feynman's variational method. The density matrix is diagonalized, and the eigenvalues and eigenfunctions are derived. These results and the calculated density matrix are used to evaluate the root-mean-square (rms) displacements R\ensuremathβ{R}_{\ensuremath{\beta}}R\ensuremathβ​ and the effective mass (EM) m∗{m}^{*}m∗ for various values of coupling strengths between electron and medium and of the temperature. We find that EM is related to the rms displacement. The temperature dependence of the EM is in qualitative agreement with experiment.
With the improvement of experimental techniques and the replacement of equipment, composition and application of solid material have been gained a series new development. So there is continuing interests in polaron. N.Tokuda discussed that the variational law of the groundstate energy, the excitedstate energies with coupling constant; Xiao Jinglin, et al. investigated that the properties of polaron in various coupling strength; Xiao Wei, et al. studied that the affact of temperature on polaron; We discussed that the properties of acoustic deformation potential polaron. This paper was going to illustrate the variational law of the effective mass of polaron with temperature and coupling strength.In this paper, the variational relation of the effective mass of polaron in crystals with temperature and coupling strength was derived by using linearcombination operator and unitary transformation method. The results indicated that the effective mass increases with increasing coupling strength when temperature is constant, resulting in the interaction of electronphonon strengthens with increasing coupling strength. The results also illustrate that the effective mass varies with variation of temperature when coupling strength is constant, the stronger coupling strength is, the more obvious the affect of temperature on effective mass. At limit of weakcoupling,the effective mass of slowmoving polaron has nothing to do with temperature because a electron is approximate a freeelectron at limit of weakcoupling and it's motion is not affected by crystals vibration.In sum, the effective mass of polaron increases with increasing coupling strength when temperature is constant, and affect of temperature on effective mass also increases with increasing coupling strength.
The approximated density matrix of the polaron system obtained in our previous paper for deriving the Feynman effective mass mF{m}_{F}mF​ and the Krivoglaz-Pekar mass mKP{m}_{\mathrm{KP}}mKP​ is reconsidered. It is shown that to obtain a consistent definition of the polaron effective mass one must impose the condition mF=mKP.{m}_{F}{=m}_{\mathrm{KP}}.mF​=mKP​. This consistent definition is based on the fact that the wave functions derived from the density matrix corresponding to each excited state must be normalized and orthogonal. This condition leads to a consistent definition of the effective mass and a different variation principle for treating excited states. Numerical results are presented.
Using the exactly soluble Feynman polaron as a point of departure, I discuss and compare the properties of three projected, or momentum dependent, partition functions of a particle coupled to an environment and reconsider the anomalies that appear in some definitions of the effective mass. Although the momentum-dependent free energy associated with the projected partition function at fixed total momentum has exponentially suppressed finite-temperature corrections and yields the correct zero-temperature effective mass, the same is not necessarily true of the free energies associated with the simpler momentum-dependent partition functions derived from either the reduced density matrix of the particle or the density matrixlike quantity obtained by replacing in the particle's nonlocal effective action the phonon Green's function with its zero-temperature form.
As ones know,the polaron is a quasiparticle formed by an electron and phonons.Due to Coulomb interaction,the electron will make the lattice deformation take place,meanwhile,the deformation potential(induced) by the lattice deformation will exert on the electron,the electron-phonon interaction influence doubtless on the state of motion of an electron,for example,the energy band and effective mass of an electron could change by the interaction,the electron can also be trapped by the deformable lattice,therefore,the research on the(polaron) is very important in understanding optical and transport properties of a solid. Studying electron-phonon interaction has been a main subject in condensed matter physics;in experiments,the influences of the temperature on the forbidden-band width of semiconductors have been observed very early,with the increase of the temperature,the forbidden-band width of most semiconductors will narrow,but a few will widen,these situations are closely relevant to the electron-phonon interaction.In particular,(recently),a large amount of experimental results,ranging from infrared spectroscopy to transport properties(involving) the colossal magnetoresistance and high-T_c superconductivity,has pointed out the importance of the electron-phonon interaction,which has renewed the interest in studying the models of the electron-phonon(coupled) system,especially,the Holstein polaron,for its relative simplicity,is the most considered model for the interaction of a single tight-binding electron coupled to phonons. In the past several decades,the Holstein poloron has been studied in detail in one-dimensional lattices,various theoretical methods,such as variational approaches,the Global-Local variational method,the(coherent)-state expansion method,have been developed to study features of the Holstein polaron including ground-state energy,polaron energy band,effective mass and so on.However,so far,the work considering the influences of the temperature on features of the Holstein polaron is little still. The main aim of this paper is to investigate the influences of the temperature on the energy-band width and the effective mass of the Holstein polaron by using the canonical transformation method.It is shown that,at certain coupling strength,the energy-band width of the Holstein polaron will narrow and the effective mass will augment with the increase of the temperature,in particular,when electron-phonon coupling strength is big enough,the energy-band width will become zero rapidly in small region of temperature,in fact,the situation is rapid transition of the polaron from the energy-band state to the self-trapping local state,this is an(interesting) result obtained in the paper.It is obvious that these features of the Holstein polaron are very useful in explaining optical and transport properties of a solid.
Adequacy of various definitions of polaron effective mass m∗{m}^{\mathrm{*}}m∗ is examined. The expression for m∗{m}^{\mathrm{*}}m∗ obtained by using the zero-temperature kernel is compared with the zero-temperature limit of the temperature-dependent effective mass obtained by employing the finite-temperature kernel. We find that consistent results are obtained when Saitoh's definition is employed. Our calculations are illustrated for a simple case of weak electron-phonon interaction.